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The problem in the image involves determining the maximum height reached by a tennis ball and the time at which it reaches this height. The mathematical expression for the height \( h(t) \) of the ball as a function of time \( t \) is given by:

\[
h(t) = 64t - 16t^2
\]

### 1. Finding the Maximum Height
The height function \( h(t) = 64t - 16t^2 \) is a quadratic equation, where the coefficient of \( t^2 \) is negative. This indicates that the parabola opens downwards, and hence, the vertex of the parabola gives the maximum height.

The time \( t \) at which the maximum height occurs is given by the formula:

\[
t = -\frac{b}{2a}
\]

Where \( a = -16 \) and \( b = 64 \).

\[
t = -\frac{64}{2(-16)} = \frac{64}{32} = 2 \text{ seconds}
\]

So, the ball reaches its maximum height at \( t = 2 \) seconds.

### 2. Maximum Height
To find the maximum height, substitute \( t = 2 \) back into the height equation:

\[
h(2) = 64(2) - 16(2)^2 = 128 - 64 = 64 \text{ meters}
\]

### Final Answers:
- **Maximum Height:** The ball reaches a maximum height of **64 meters**.
- **Time to reach Maximum Height:** The ball reaches this height at **2 seconds**.

Would you like more details on any of these steps or have any other questions?

### Related Questions:
1. How would the maximum height change if the initial velocity were doubled?
2. What would the time to reach maximum height be if the acceleration due to gravity were different?
3. Can you derive the equation \( h(t) \) from the basic principles of physics?
4. What is the total time the ball spends in the air before it hits the ground?
5. How would the problem change if the ball was launched from a different height?

### Tip:
When dealing with quadratic equations in physics, the vertex of the parabola often provides key information, such as the maximum or minimum value depending on the context.

Learn how to find the maximum height reached by a tennis ball using a quadratic equation. Explore the vertex formula for quadratic equations and calculate the maximum height and time to reach it. Suitable for Grades 10-12.

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